1. **Determine if each number is rational or irrational:** a) $\sqrt{\frac{49}{16}}$ - Step 1: Simplify the square root of a fraction using $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. $$\sqrt{\frac{49}{16}} = \frac{\sqrt{49}}{\sqrt{16}}$$ - Step 2: Calculate the square roots. $$\frac{7}{4}$$ - Step 3: Since $\frac{7}{4}$ is a ratio of two integers, it is a rational number. b) $\sqrt[3]{-30}$ - Step 1: Cube roots of negative numbers are defined and equal to the negative of the cube root of the positive number. $$\sqrt[3]{-30} = -\sqrt[3]{30}$$ - Step 2: Since 30 is not a perfect cube, $\sqrt[3]{30}$ is irrational. - Step 3: Therefore, $\sqrt[3]{-30}$ is irrational. c) $1.21$ - Step 1: $1.21$ is a terminating decimal. - Step 2: All terminating decimals are rational numbers because they can be expressed as fractions. - Step 3: $1.21 = \frac{121}{100}$, so it is rational. 2. **Order the numbers from least to greatest:** $\sqrt{2}$, $\sqrt[3]{-2}$, $\sqrt{6}$, $\sqrt{11}$, $\sqrt{30}$ - Step 1: Approximate each value. $$\sqrt{2} \approx 1.414$$ $$\sqrt[3]{-2} \approx -1.26$$ $$\sqrt{6} \approx 2.449$$ $$\sqrt{11} \approx 3.317$$ $$\sqrt{30} \approx 5.477$$ - Step 2: Order from least to greatest: $$\sqrt[3]{-2} < \sqrt{2} < \sqrt{6} < \sqrt{11} < \sqrt{30}$$ 3. **Simplify each radical:** a) $\sqrt{63}$ - Step 1: Factor 63 into $9 \times 7$. - Step 2: Use $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. $$\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}$$ b) $\sqrt[3]{108}$ - Step 1: Factor 108 into $27 \times 4$. - Step 2: Use $\sqrt[3]{ab} = \sqrt[3]{a} \times \sqrt[3]{b}$. $$\sqrt[3]{108} = \sqrt[3]{27 \times 4} = \sqrt[3]{27} \times \sqrt[3]{4} = 3\sqrt[3]{4}$$ c) $\sqrt[4]{128}$ - Step 1: Factor 128 into $16 \times 8$. - Step 2: Use $\sqrt[4]{ab} = \sqrt[4]{a} \times \sqrt[4]{b}$. $$\sqrt[4]{128} = \sqrt[4]{16 \times 8} = \sqrt[4]{16} \times \sqrt[4]{8} = 2\sqrt[4]{8}$$ 4. **Write each radical in simplest form if possible:** a) $\sqrt{30}$ - Step 1: 30 factors into $2 \times 3 \times 5$, no perfect square factors. - Step 2: So, $\sqrt{30}$ is already in simplest form. b) $\sqrt[3]{32}$ - Step 1: 32 is $2^5$. - Step 2: $\sqrt[3]{32} = \sqrt[3]{2^5} = 2^{5/3} = 2^{1 + 2/3} = 2 \times 2^{2/3} = 2\sqrt[3]{4}$ c) $\sqrt[4]{48}$ - Step 1: 48 factors into $16 \times 3$. - Step 2: $\sqrt[4]{48} = \sqrt[4]{16 \times 3} = \sqrt[4]{16} \times \sqrt[4]{3} = 2 \sqrt[4]{3}$ 5. **Write each mixed radical as an entire radical:** a) $7\sqrt{3}$ - Step 1: Write 7 as $\sqrt{49}$. - Step 2: $7\sqrt{3} = \sqrt{49} \times \sqrt{3} = \sqrt{49 \times 3} = \sqrt{147}$ b) $2\sqrt[3]{4}$ - Step 1: Write 2 as $\sqrt[3]{8}$. - Step 2: $2\sqrt[3]{4} = \sqrt[3]{8} \times \sqrt[3]{4} = \sqrt[3]{8 \times 4} = \sqrt[3]{32}$ c) $2^{5}\sqrt[5]{3}$ - Step 1: Write $2^{5}$ as $\sqrt[5]{2^{25}}$ because $\sqrt[5]{2^{25}} = 2^{25/5} = 2^{5}$. - Step 2: Multiply inside the radical: $$2^{5}\sqrt[5]{3} = \sqrt[5]{2^{25}} \times \sqrt[5]{3} = \sqrt[5]{2^{25} \times 3} = \sqrt[5]{3 \times 2^{25}}$$ 6. **Simplify and evaluate without a calculator:** a) $1000^{1/3}$ - Step 1: $1000 = 10^3$ - Step 2: $1000^{1/3} = (10^3)^{1/3} = 10^{3 \times \frac{1}{3}} = 10^1 = 10$ b) $0.25^{1/2}$ - Step 1: $0.25 = \frac{1}{4} = (\frac{1}{2})^2$ - Step 2: $0.25^{1/2} = \left((\frac{1}{2})^2\right)^{1/2} = (\frac{1}{2})^{2 \times \frac{1}{2}} = \frac{1}{2}$ c) $(-8)^{1/3}$ - Step 1: Cube root of -8 is -2 because $(-2)^3 = -8$ - Step 2: $(-8)^{1/3} = -2$ d) $\left(\frac{16}{81}\right)^{1/4}$ - Step 1: $16 = 2^4$, $81 = 3^4$ - Step 2: $\left(\frac{16}{81}\right)^{1/4} = \frac{16^{1/4}}{81^{1/4}} = \frac{2}{3}$ 7. **Write $26^{5/2}$ in radical form in 2 ways:** - Way 1: $26^{5/2} = \left(26^{1/2}\right)^5 = \left(\sqrt{26}\right)^5$ - Way 2: $26^{5/2} = \sqrt{26^5}$ 8. **Write $\sqrt{6}$ and $\left(\sqrt[4]{19}\right)^3$ in exponent form:** - $\sqrt{6} = 6^{1/2}$ - $\left(\sqrt[4]{19}\right)^3 = \left(19^{1/4}\right)^3 = 19^{3/4}$ 9. **Evaluate:** a) $0.01^{3/2}$ - Step 1: $0.01 = 10^{-2}$ - Step 2: $0.01^{3/2} = (10^{-2})^{3/2} = 10^{-3} = 0.001$ b) $(-27)^{4/3}$ - Step 1: $(-27)^{4/3} = \left((-27)^{1/3}\right)^4$ - Step 2: Cube root of -27 is -3 - Step 3: $(-3)^4 = 81$ c) $81^{3/4}$ - Step 1: $81 = 3^4$ - Step 2: $81^{3/4} = (3^4)^{3/4} = 3^{4 \times \frac{3}{4}} = 3^3 = 27$ d) $0.75^{1.2}$ - Step 1: Approximate $0.75^{1.2}$ using logarithms or calculator (approximate value) - Step 2: $0.75^{1.2} \approx 0.68$ 10. **Use formula $b = 0.01 m^{2/3}$ to estimate brain mass:** a) Moose with $m=512$ kg - Step 1: Calculate $512^{2/3}$ - Step 2: $512 = 8^3$, so $512^{1/3} = 8$ - Step 3: $512^{2/3} = (512^{1/3})^2 = 8^2 = 64$ - Step 4: $b = 0.01 \times 64 = 0.64$ b) Cat with $m=5$ kg - Step 1: Calculate $5^{2/3}$ - Step 2: $5^{1/3} \approx 1.71$ - Step 3: $5^{2/3} = (5^{1/3})^2 \approx 1.71^2 = 2.92$ - Step 4: $b = 0.01 \times 2.92 = 0.0292$ Final answers summarized: - 1a) Rational - 1b) Irrational - 1c) Rational - 2) $\sqrt[3]{-2} < \sqrt{2} < \sqrt{6} < \sqrt{11} < \sqrt{30}$ - 3a) $3\sqrt{7}$ - 3b) $3\sqrt[3]{4}$ - 3c) $2\sqrt[4]{8}$ - 4a) $\sqrt{30}$ (simplest) - 4b) $2\sqrt[3]{4}$ - 4c) $2\sqrt[4]{3}$ - 5a) $\sqrt{147}$ - 5b) $\sqrt[3]{32}$ - 5c) $\sqrt[5]{3 \times 2^{25}}$ - 6a) 10 - 6b) $\frac{1}{2}$ - 6c) -2 - 6d) $\frac{2}{3}$ - 7) $\left(\sqrt{26}\right)^5$ or $\sqrt{26^5}$ - 8) $6^{1/2}$ and $19^{3/4}$ - 9a) 0.001 - 9b) 81 - 9c) 27 - 9d) $\approx 0.68$ - 10a) 0.64 - 10b) 0.0292